This is my first assignment on these, so I would greatly appreciate your help.
Translate the following into predicate calculus. For each answer, also state the assumed universe of discourse.
a) "Anyone who was an ancient Roman citizen and tried to kill Caesar was not loyal to Caesar."
My attempt:
Ex(R(x) & k(x)) -> ~L(x)
R(x) = x is a Roman citizen, k(x) = x is tried to kill Caesar, L(x) = x is loyal to Caesar
b) "All cats which are calico, are female."
My attempt:
Ax(C(x) & O(x)) -> F(x)
C(x) = x is a cat, O(x) = x is calico, F(x) = x is female
c) "Some Texans have never left the state of Texas."
My attempt:
~Ax(T(x) & L(x))
T(x) = x is a Texan, L(x) = x is left Texas
Is the universe of discourse kind of like the "key" for the variables, as I included?
a), b) are correct, although I don't see that you have stated the universe of discourse for any of the three sentences.
However, but c) is not correct. The sentence you have for c) is close but not quite right: moving the negation inward shows that it's equivalent to: $$\begin{align} \neg\forall x\,(T(x)\land L(x)) &\equiv \exists x\,\neg\,(T(x)\land L(x)) \\ &\equiv \exists x\,(T(x)\to \neg L(x)) \\ &\equiv \exists x\,(\neg T(x)\lor \neg L(x)), \end{align}$$ which says "Some people either aren't Texans or have never left Texas", assuming $x$ ranges over only people (i.e. the universe consists only of people).
c) should be: $$ \exists x\, (T(x) \land \neg L(x)), $$ where $T(x) := \text{$x$ is a Texan}$, $L(x) := \text{$x$ has left Texas (at some point)}$.
The universe of discourse for c) is all people (all living people, let's say).
In a), your predicates seem to be set in the present tense, and you do filter for "is a(n ancient) Roman citizen", so your universe of discourse can be all contemporaries of Caesar. In fact, it could even be *anything contemporaneous with Caesar", since presumably contemporaneous cats, kitchen utensils, and so on were not Roman citizens.
In b), few restrictions apply. $x$ can range over everything — cats, dogs, people, hydrogen atoms, the integers, etc. — and the statement remains true because $C(x)$ is in the antecedent.